# Dealing with constraints

#### Fabrice Zaoui

#### October 05 2018

# Short Description

caRamel is a multiobjective evolutionary algorithm combining the MEAS algorithm and the NGSA-II algorithm.

Download the package from CRAN or GitHub and then install and load it.

Dealing with constraints is possible with caRamel by returning a NaN value for an infeasible solution. See the example below.

    library(caRamel)

# Test functions

## Constr-Ex problem

Constr-Ex test function has two objectives with two variables and two inequality constraints.

    constr_ex <- function(i) {
      # functions f1 and f2
      s1 <- x[i,1]
      s2 <- (1. + x[i,2]) / x[i,1]
      # now test for the feasibility
      # constraint g1
      if((x[i,2] + 9. * x[i,1] - 6.) < 0. | (-x[i,2] + 9. * x[i,1] -1.) < 0.) {
        s1 <- NaN
        s2 <- NaN
      }
      return(c(s1, s2))
    }

Note that :

- parameter i is mandatory for the management of parallelism.
- the variable must be named x and is a matrix of size [npopulation, nvariables].

The variable lies in the range [0.1, 1] and [0, 5]:

    nvar <- 2 # number of variables
    bounds <- matrix(data = 0., nrow = nvar, ncol = 2) # upper and lower bounds
    bounds[1, 1] <- 0.1
    bounds[1, 2] <- 1.
    bounds[2, 1] <- 0.
    bounds[2, 2] <- 5.

Both functions are to be minimized:

    nobj <- 2 # number of objectives
    minmax <- c(FALSE, FALSE) # min and min

Before calling caRamel in order to optimize the Constr_Ex problem, some algorithmic parameters need to be set:

    popsize <- 100 # size of the genetic population
    archsize <- 100 # size of the archive for the Pareto front
    maxrun <- 1000 # maximum number of calls
    prec <- matrix(1.e-3, nrow = 1, ncol = nobj) # accuracy for the convergence phase

Then the minimization problem can be launched:

    results <-
      caRamel(nobj,
              nvar,
              minmax,
              bounds,
              constr_ex,
              popsize,
              archsize,
              maxrun,
              prec,
              carallel=FALSE) # no parallelism

    ## Beginning of caRamel optimization <-- Tue May 12 11:01:20 2026

    ## Number of variables : 2

    ## Number of functions : 2

    ## Done in 1.63086438179016 secs --> Tue May 12 11:01:22 2026

    ## Size of the Pareto front : 98

    ## Number of calls : 1000

Test if the convergence is successful:

    print(results$success==TRUE)

    ## [1] TRUE

Plot the Pareto front:

    plot(results$objectives[,1], results$objectives[,2], main="Constr_Ex Pareto front", xlab="Objective #1", ylab="Objective #2")

    plot(results$parameters, main="Corresponding values for X", xlab="Element of the archive", ylab="X Variable")